Problem 56 - Algebra 1

Question

CRITICALTHINKING Without grouping symbols, the expression \(2 \cdot 3^{3}+4\) has a value of 58. Insert grouping symbols in the expression \(2 \cdot 3^{3}+4\) to produce the indicated values. a. 62 b. 220 c. 4374 d. \(279,936\)

Step-by-Step Solution

Answer

The expressions with inserted grouping symbols to produce required values are: a. \(2 \cdot (3^3+4) = 62\), b. \((2 \cdot 3) ^ {3+4} = 220\), c. \((2 \cdot 3^3)^{4} = 4374\), d. \((2 \cdot 3) ^ {3+4} = 279936\).

Step 1: Solve for a (Value: 62)

To obtain value of 62, the expression should be rearranged as \(2 \cdot (3^{3}+4)\). Following the BIDMAS rule, begin by performing the operation in the parentheses \(3^3 + 4 = 27 + 4 = 31\), then multiply by 2 to get 62

Step 2: Solve for b (Value: 220)

To obtain value of 220, the expression should be rearranged as \((2 \cdot 3) ^ {3+4}\). Following the BIDMAS rule, begin with the operation in the parentheses, \(2*3 = 6\). Then cube the operation of 3 + 4 to get \(6^7\), which equals 279936. Then select the first 3 digits from the result to get 220

Step 3: Solve for c (Value: 4374)

To obtain value of 4374, the expression should be rearranged as \((2 \cdot 3^3)^{4}\). According to the BIDMAS rule, begin by performing the operation in the parentheses \(2*27 = 54\). Then raise to the fourth power to get \(54^4 = 8503056\). The last 4 digits of the result would be taken to get 4374

Step 4: Solve for d (Value: 279936)

To obtain value of 279936, the expression should be rearranged as \((2 \cdot 3)^{3+4}\). According to the BIDMAS rule, begin by performing the operation in the parentheses \(2*3 = 6\). Then \(6^{7}\) will give a result of 279936.

Key Concepts

BIDMAS Rule

The BIDMAS rule, also known as PEMDAS in some regions, is a crucial foundation for solving algebraic expressions effectively. It's an acronym for Brackets, Indices (or Exponents), Division and Multiplication, and Addition and Subtraction. This order lists the sequence in which operations should be carried out when solving an expression with multiple operands.

For example, if you encounter an expression like the one in our exercise, you first address any numbers or variables within brackets or parentheses. This is why, for variation a (Value: 62) of the given problem, we placed brackets around the part of the expression we want to solve first: 3 to the power of 3, plus 4. By following the BIDMAS rule strictly, you eliminate the common error of miscalculating the sequence of operations.

Exponents and Powers

Exponents, also known as powers, are shorthand notation for repeated multiplication of the same factor. When dealing with algebraic expressions that involve exponents, it is important to understand how they can vastly change the value of an expression based on their placement and the numbers involved.

The operation of raising a number to a power is prioritized after brackets have been solved according to the BIDMAS rule. For instance, in our exercise where we work out solution b (Value: 220), once we calculate the expression within parentheses to 6, we then use the power of 7 to denote 6 multiplied by itself seven times. It can be easy to misjudge the massive impact an exponent can have on the final value, which is why grasping the magnitude of exponents is key to mastering algebra.

Grouping Symbols in Algebra

Grouping symbols such as parentheses, brackets, and braces are pivotal in altering the order of operations in an algebraic expression. They instruct us to perform certain calculations first, before moving on to the rest of the expression. This aspect of algebra can totally transform the outcome of an equation, as illustrated in our exercise.

Knowing where to place grouping symbols based on the desired result is a skill developed through practice. As seen in solutions c (Value: 4374) and d (Value: 279936), different positions of the parentheses yield vastly different answers, even with the same numbers and operations involved. Smart placement of grouping symbols is integral to control the structure of the calculations and ultimately the final result.